Reverse time migration image improvementusing integral transforms

Juan Guillermo Paniagua C.

M.Sc. in EngineeringPh.D. student in Mathematical Engineering

GRIMMAT - Research group in mathematical modeling

Advisor: Olga Lucıa Quintero M.

2/40

Content

Integral transforms

Problem statement

Laguerre-Gauss transform

Numerical implementations

Future work

References

3/40

Integral transforms

An integral transform is a map from one function f (x), to another, F(k)(Deakin,1985). The integral transform of a function f (x) is denoted byT {f (x)} = F(k), and defined by

F(k) =

∫S

K (x, k)f (x)dk (1)

wherex = (x1, x2, ...., xn)k = (k1, k2, ...., kn)K (x, k)= Kernel of the transformS ⊂ Rn

4/40

Integral transforms

Some integral transforms

Fourier transform F{f (x)} = F(k) = 1(2π)n/2

∫ ∞−∞· · ·∫ ∞−∞

f (x)e−i(k·x)dx

Mellin transform M−{f (r)} = F−(p) =

∫ ∞a

(rp−1 − a2p

rp+1

)f (r)dr

Hankel transfom Hn =

∫ ∞0

rJn(kr)f (r)dr

Radon transform f (p, u) = R{f (x)} =∫ ∞−∞

f (x)δ(p − x · u)dx

5/40

Integral transforms

All integral transforms presented above are linear integral transforms, i.e,∀α, β ∈ R, ∀x ∈ Rn and f , g functions on Rn:

T {αf (x) + βg(x)} =

∫S

K (x, k)(αf (x) + βg(x))dk

= αT {f (x)}+ βT {g(x)}

6/40

Some areas of application of integral transfoms

I Fluid mechanics.

I Signal and image processing.

I Quantum mechanics.I Geophysics.

I One way wave equation migration (OWWE).I Phase shift migration.I PSPI and SS migration.

I Kirchhoff migration.

7/40

Problem statement

Reverse time migration (RTM)

8/40

Problem statement

Reverse time migration (RTM)Acoustic wave equation

1

c(x, z)2

∂2u(x, z, t)

∂t2−∇2u(x, z, t) = s(x, z, t)

1. Forward propagation of the sourcewavefield.

2. Backward propagation of the re-ceivers wavefield.

3. Apply a criterion to construct theseismic image (Imaging condition).

9/40

Problem statement

Cross correlation imaging condition

ICC(x, z) =smax∑j=1

tmax∑i=1

S(x, z; ti ; sj)R(x, z; ti ; sj)

whereS: Source wavefieldR: Receiver wavefieldz: Depthx : Distancet : Time

10/40

Problem statement

Velocity model of two layers synthetic model

11/40

Problem statement

Cross correlation image of two layers model

12/40

Problem statement

Velocity model 2D SEG/EAGE salt model

13/40

Problem statement

Cross correlation image of 2D SEG/EAGE salt model

14/40

Methods to eliminate the artifacts

1. Wavefield propagation approches (Loewenthal, 1983, 1987, Baysal,1984, Fletcher, 2005).

2. Imaging condition approches (Valenciano and Biondi, 2003 Kaelin etal, 2006, Guitton, 2007, Liu, 2011, Whitmore, 2012, Pestana et al,2013, Shragge, 2014).

3. Post-imaging condition approches (Youn, 2001, Guitton et al, 2006).

15/40

Imaging condition approches

Source illumination imagingcondition

Isill(x, z) =

∑t S(x, z, t)R(x, z, t)∑

t S2(x, z, t)

Receiver illumination imagingcondition

Irill(x, z) =

∑t S(x, z, t)R(x, z, t)∑

t R2(x, z, t)

16/40

Imaging condition approches

Inverse scattering imaging condition

IIS(x, z) =Tmax∑t=0

[∇S(x, z, t) · ∇R(x, z, t)− 1

c(x, z)2

∂S(x, z, t)

∂t

∂R(x, z, t)

∂t

]

17/40

Imaging condition approches

Impedance sensitivity kernel imaging condition

Ik(x, z) =1

v2(x, z)

∫∂

∂tPF (x, z, t)

∂

∂tPB(x, z, t)dt+

∫∇PF (x, z, t) · ∇PB(x, z, t)dt

18/40

Post-imaging condition approches

Laplacian filtering

ILP(x, z) =∂

∂x2Icc(x, z) +

∂

∂z2Icc(x, z)

Icc(x, z): Cross correlation image.

19/40

Laguerre-Gauss transform

The Laguerre-Gauss transform of I(x, y) is given by (Wang et al, 2006, Guoet al, 2006):

I(x, y) =

∫ ∞−∞

∫ ∞−∞

LG(fx , fy)I(fx , fy)e2πi(fx x,fy y)dfx dfy (2)

where

LG(fx , fy) = (fx + ify)e−(f 2

x +f 2y )/ω

2= ρe−(ρ

2/ω2)eiβ (3)

ρ =√

f 2x + f 2

y , β = tan−1(

fyfx

)are the polar coordinates in the spatial

frequency domain.

20/40

Laguerre-Gauss transform

I(x, y) = |I(x, y)|eiθ(x,y) = I(x, y) ∗ LG(x, y)

From equation (3) we obtain

LG(x, y) = F−1{LG(fx , fy)} = (iπ2ω4)(x + iy)e−π2ω2(x2+y2)

= (iπ2ω4)[re−π2r2ω2

eiα]

where r =√

x2 + y2, α = tan−1( y

x

)are the spatial polar coordinates.

21/40

Laguerre-Gauss transform

Spiral phase function Toroidal amplitud

Figure: Laguerre Gauss Filter (Wang et al, 2006)

22/40

Two layers synthetic model

Velocity model Cross correlation image

23/40

Two layers synthetic model

Laplacian image Laguerre Gauss image

24/40

2D SEG/EAGE salt model

Velocity model Cross correlation image

25/40

2D SEG/EAGE salt model

Cross correlation image Scaled Cross correlation image

26/40

2D SEG/EAGE salt model

Laplacian image

27/40

2D SEG/EAGE salt model

Laguerre-Gauss image

28/40

29/40

Future work

I Write a paper reporting the results obtained with this post-imagingcondition.

I Report the advances obtained in the imaging condition to ICP.

I Use computer cluster to compute the RTM algorithms parallelized.

I Measure the accuracy of the image obtained by Laguerre-Gauss Fil-tering compared with the true image (velocity model).

I Implement Laguerre-Gauss transform to modify or propose a new imag-ing condition.

30/40

References

[1] Arntsen, B., Kritski, B., Ursin, B., and Amundsen, L., 2013, Shot-profile amplitude crosscorrelation imaging condition: Geophysics, 78,4, S221 - S231.

[2] Baysal, E., Kosloff, D. D., and Sherwood, J. W. C., 1983, Reversetime migration: Geophysics, 48,1514 -1524.

[3] Baysal, E., Kosloff, D. D., and Sherwood, J. W. C., 1984, A two waynonreflecting wave equation: Geophysics, 49, 2, 132 -141.

[4] Bokor, N. and Iketaki, Y., 2009, Laguerre-Gaussian radial Hilberttransform for edge-enhancement Fourier transform x-ray microscopy:Optics Express, 19, 7, 5533 -5539.

30/40

References

[5] Chattopadhyay, S., and McMechan, G., 2008, Imaging conditions forprestack reverse time migration: Geophysics, 73, 3, S81 - S89.

[6] Claerbout, J. F., 1971, Toward a unified theory of reflector mapping:Geophysics, 30, 467 - 481.

[7] Claerbout, J. F., 1985, Imaging the Earth’s interior: Blackwell Scien-tific Publications.

[8] Costa, J., Silva, F., Alcantara, R., Schleicher, J., and Novais, A.,2009, Obliquity-correction imaging condition for reverse time migra-tion: Geophysics, 74, 3, S57 - S66.

31/40

References

[9] Davis, J., McNamara, D., and Cottrell, D., 2000, Image processingwith the radial Hilbert transform: theory and experiments: Optics Let-ters, 25, 2, 99 - 101.

[10] Deakin, M., 1985, Euler’s invention of integral transforms: Archive forHistory of Exact Sciences, 33, 4, 307 - 319.

[11] Debnath, L., and Bhatta, D. 2010, Integral transforms and their appli-cations. CRC press.

[12] Fletcher, R., Fowler, P., and Kitchenside, P., 2005, Suppressing ar-tifacts in prestack reverse time migration: 75th International AnnualMeeting, SEG, Expanded abstracts, 2049 - 2051.

32/40

References

[13] Forel, D., Benz, T., and Pennington, W., 2005, Seismic data process-ing with Seismic Unix. A 2D Seismic data processing primer. Courseseries No. 12: Society of exploration Geophysicist.

[14] Freund, I., and Freilikher, V. 1997, Parameterization of anisotropicvortices. JOSA A, 14, 8, 1902 - 1910.

[15] Guitton, A., Kaelin, B., and Biondi, B., 2006, Least-square attenuationof reverse time migration: 76th International Annual Meeting, SEG,Expanded abstracts, 2348 - 2352

[16] Guitton, A., Valenciano, A., Bevc, D., and Claerbout, J., 2007,Smoothing imaging condition for shot-profile migration: Geophysics,72, 3, 149 - 154.

33/40

References

[17] Gou, C., Han, Y., and Xu, J., 2006, Radial Hilbert transform withLaguerre-Gaussian spatial filters: Optics Letters, 31, 10, 1394 - 1396.

[18] Haney, M., Bartel, L., Aldridge, D., and Symons, N., 2005, Insight intothe output of reverse time migration: What do the amplitudes mean?:75th International Annual Meeting, SEG, Expanded abstracts, 1950- 1953.

[19] Hu, L., McMechan, G., 1987, Wave-field transformations of verticalseismic profiles: Geophysics, 52, 307 - 321.

[20] Kaelin, B. and Guitton, A., 2006, Imaging condition for reverse timemigration: 76th International Annual Meeting and exposition, SEG,Expanded abstracts, 2594 - 2598

34/40

References

[21] Kosloff, D., Baysal, E., 1983, Migration with the full wave equation:Geophysics, 48, 677 - 687.

[22] Liu, F., Zhang, G., Morton, S., and Leveille, J., 2011, An effectiveimaging condition for reverse time migration using wavefield decom-position: Geophysics, 76, 1, 29 - 39.

[23] Loewenthal, D., Mufti, I., 1983, Reverse time migration in spatial fre-quency domain: Geophysics, 48, 5, 627 - 635.

[24] Loewenthal, D., Stoffa, P. and Faria, E., 1987, Suppressing the un-wanted reflections of the full wave equation: Geophysics, 52, 7, 1007- 1012.

35/40

References

[25] Luo, Y., Zhu, H., Nissen-Meyer, T., Morency, C., and Tromp, J., 2009,Seismic modeling and imaging based upon spectral-element and ad-joint methods: The Leading edge, 28, 568 - 574.

[26] Macdonald, J. R., and Brachman, M. K., 1956, Linear-system integraltransform relations. Reviews of modern physics, 28, 4, 393 - 422.

[27] McMechan, G, A, 1983, Migration by extrapolation of time - dependboundary values: Geophysics Prospecting, 31, 413 - 420.

[28] Nguyen, B., McMechan, G., 2013, Excitation amplitude imaging con-dition for prestack reverse time migration: Geophysics, 78, 1, 37 -46.

36/40

References

[29] Pratt, W. K., 2001, Digital image processing: Wiley Interscience.

[30] Pestana, R., and Dos Santos, A., 2013, RTM imaging condition usingimpedance sensitivity kernel combinated with the Poynting vector:13th International Congress of the Brazilian Geophysical Society, 1 -5.

[31] Schleicher, J., Costa, J., Novais, A., 2007, A comparison of imagingfor wave-equation shot-profile migration: Geophysics, 73, 6, S219 -S227

[32] Shragge, J., 2012, Reverse time migration from topography: Geo-physics, 79, 4, 1-12.

37/40

References

[33] Stolk, C., De Hoop, M., and Root, T., 2009, Linearized inverse scat-tering based on seismic Reverse Time Migration: Proceedings of theProject Review, Geo-Mathematical imaging group, 1, 91 - 108.

[34] Valenciano, A., and Biondi, B., 2003, 2D Deconvolution imaging con-dition for shot profile migration: 73th International Annual Meetingand exposition, SEG, Expanded abstracts, 1059 - 1062

[35] Vivas, F., and Pestana, R., 2007, Imaging condition to true amplitudeshot-profile migration: A comparison of stabilization techniques: 10thInternational congress ofnthe Barazilian Geophysical Society, 1668 -1672.

38/40

References

[36] Wang, W., Yokozeki, T., Ishijima, R., Takeda, M., and Hanson, S.G. 2006, Optical vortex metrology based on the core structures ofphase singularities in Laguerre-Gauss transform of a speckle pattern.Optics Express, 14, 22, 10195 - 10206.

[37] Wang, W., Yokozeki, T., Ishijima, R., Wada, A., Miyamoto, Y., Takeda,M., and Hanson, S. G. 2006, Optical vortex metrology for nanomet-ric speckle displacement measurement. Optics express, 14. 1, 120 -127.

[38] Whitmore, N., and Crawley, S., 2012, Applications of RTM inversescattering imaging conditions: 82nd Annual International Meeting,SEG, Expanded abstracts, 779 . 784.

39/40

References

[39] Yoon, K., Marfurt, K., 2006, Reverse time migration using the Poynt-ing vector: Exploration Geophysics, 37, 102 - 107.

[40] Youn, O., Zhou, H., 2001, Depth imaging with multiples: Geophysics,66, 1, 246 - 255.

40/40